Let be a commutative Noetherian ring of dimension and a commutative cancellative torsion-free seminormal monoid. Then: (1) Let be a ring of type and be a projective -module of rank . Then the action of on is transitive and (2) Assume is a regular local ring containing a field such that either or and -. Let be a ring of type and be a regular parameter. Then all finitely generated projective modules over , and are free. When is free both results are due to Keshari and Lokhande (2014).
"Unimodular rows over monoid extensions of overrings of polynomial rings." J. Commut. Algebra 14 (4) 583 - 589, Winter 2022. https://doi.org/10.1216/jca.2022.14.583